Suppose that the Dedekind conjecture is true for \( K \) , that is suppose that \( L(s,\chi _1^{\prime }) \)  is entire. Observe that we can replace \( \theta _3 \)  with \( \theta _1 + \theta _2 \)  in (REF ) and Theorem REF  would still be true. Now suppose that \( \chi _1^{\prime } \)  is a direct sum of monomial representations. Then \( L(s,\chi _1) \)  can be written as a product of Hecke \( L \)  functions. Now, a recent result of Bourgain [1]} states that \( \theta _1 \)  can be taken to be equal to \( 13/42 \)  and using the subconvexity results of Söhne [2]}, we can take \( \theta _2 \)  as \( n^{\prime \prime }/3 - 1 \) . Thus in this case we can see that \( \theta _1 + \theta _2 < \theta _3 \)  and thus this would lead to a larger value for \( \delta \)  and in turn a sharper error estimate in Theorem REF .
